(i) By adding 0 to the \((n+1)\) -th position of every codeword of \(\mathcal {C}\) , one gets an \([n+1,k,d]_q\) -linear code. For the \((n+1)\) -th position, we have a recovery set \(\lbrace n+1\rbrace \) . Thus, the new code has locality \(r\)  as well.
(ii) Let \(H\)  be a parity-check matrix of \(\mathcal {C}\) . From [1]}, there are \(d\)  columns of \(H\)  which are linearly dependent. Without loss of generality, assume that the last column of \(H\)  is not in these \(d\)  columns. We delete the last column of \(H\)  to form a \((n-k)\times (n-1)\)  matrix \(H_1\) . Let \(\mathcal {C}_1\)  be the code with \(H_1\)  as its parity-check matrix. It is clear that \(\mathcal {C}_1\)  is an \([n-1,\geqslant k-1, d]_q\) -linear code. To obtain the locality, for each \(i\in [n-1]\) , there exists a codeword \({\bf u}\in \mathcal {C}^\perp \)  such that \(i\in {\rm supp }({\bf u})\)  and \(|{\rm supp }({\bf u})|\leqslant r+1\) . Let \({\bf u}_1\)  be the vector obtained from \({\bf u}\)  by deleting the last position. Then \({\bf u}_1\in \mathcal {C}_1^\perp \) , \(i\in {\rm supp }({\bf u}_1)\)  and \(|{\rm supp }({\bf u}_1)|\leqslant r+1\) . This implies that the position \(i\)  has a recovery set of size at most \(r+1\) . Hence, \(\mathcal {C}_1\)  is an \([n-1,\geqslant k-1,d;r]_q\) -locally repairable code.
(iii) Without loss of generality, we may assume that \(\lbrace n-t+1,n-t+2,\dots ,n\rbrace \)  is a recovery set of size \(t\) . Let \(H\)  be a parity-check matrix of \(\mathcal {C}\) . We delete the last \(t-s\)  columns of \(H\)  to form an \((n-k)\times (n-t+s)\)  matrix \(H_2\) . Let \(\mathcal {C}_2\)  be the code with \(H_2\)  as its parity-check matrix. It is clear that \(\mathcal {C}_2\)  is an \([n-t+s,\geqslant k-t+s,\geqslant d]_q\) -linear code. As in (ii), we can show that \(\mathcal {C}_2\)  has locality \(r\)  as well. Furthermore, \(\lbrace n-t+1,n-t+2,\cdots , n-t+s\rbrace \)  is a recovery set that is disjoint with other recovery sets. Now we delete the last \(s\)  positions of \(\mathcal {C}_2\)  to obtain \(\mathcal {C}_3\) . It is easy to see that \(\mathcal {C}_3\)  is an \([n-t,\geqslant k-t+s,\geqslant d-s]_q\) -linear code. As a whole recovery set of \(\mathcal {C}_2\)  is deleted, \(\mathcal {C}_3\)  has disjoint recovery sets with each size being at most \(r+1\) , i.e., \(\mathcal {C}_3\)  has locality \(r\) .
